Competing orders: beyond Landau-Ginzburg-Wilson theory

1
Competing orders
beyond Landau-Ginzburg-Wilson theory
Colloquium article in Reviews of Modern Physics
75, 913 (2003)
Leon Balents (UCSB) Lorenz Bartosch (Yale)
Anton Burkov (UCSB) Eugene Demler (Harvard)
Matthew Fisher (UCSB) Anatoli Polkovnikov
(Harvard) Krishnendu Sengupta (Yale)
T. Senthil (MIT) Ashvin
Vishwanath (MIT) Matthias Vojta (Karlsruhe)
Talk online Google Sachdev
2
Competing orders
beyond Landau-Ginzburg-Wilson theory
Colloquium article in Reviews of Modern Physics
75, 913 (2003)
Leon Balents (UCSB) Lorenz Bartosch (Yale)
Anton Burkov (UCSB) Eugene Demler (Harvard)
Matthew Fisher (UCSB) Anatoli Polkovnikov
(Harvard) Krishnendu Sengupta (Yale)
T. Senthil (MIT) Ashvin
Vishwanath (MIT) Matthias Vojta (Karlsruhe)
Talk online Google Sachdev
3
Possible origins of the pseudogap in the cuprate
superconductors

• Phase fluctuations, preformed pairs
  Complex order parameter Ysc
• Charge/valence-bond/pair-density/stripe
  order Order parameters
  (density r represents any
  observable invariant under spin rotations,
  time-reversal, and spatial inversion)
  
• Spin liquid

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Order parameters are not independent
Ginzburg-Landau-Wilson approach to competing
order parameters
Distinct symmetries of order parameters permit
couplings only between their energy densities
S. Sachdev and E. Demler, Phys. Rev. B 69, 144504
(2004).
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Predictions of LGW theory
First order transition
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Predictions of LGW theory
First order transition
7

• Non-superconducting quantum phase must have some
  other order
• Charge order in an insulator
• Fermi surface in a metal
• Topological order in a spin liquid

This requirement is not captured by LGW theory.
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Outline

1. Superfluid-insulator transitions of bosons on
   the square lattice at fractional filling Dual
   vortex theory and the magnetic space group.
2. Application to a short-range pairing model for
   the cuprate superconductors Charge order and
   d-wave superconductivity in an effective theory
   for the spin S0 sector.
3. Implications for STM

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A. Superfluid-insulator transitions of bosons
on the square lattice at fractional filling
Dual vortex theory and
the magnetic space group.
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Bosons at density f 1
LGW theory continuous quantum transitions
between these states
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Bosons at density f 1/2
Weak interactions superfluidity
Strong interactions Candidate insulating states
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys.
Rev. B 63, 134510 (2001) S. Sachdev and K. Park,
Annals of Physics, 298, 58 (2002)
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Predictions of LGW theory
First order transition
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Superfluid insulator transition of hard core
bosons at f1/2
A. W. Sandvik, S. Daul, R. R. P. Singh, and D.
J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)
Large scale (gt 8000 sites) numerical study of the
destruction of superfluid order at half filling
with full square lattice symmetry
g
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Boson-vortex duality
Quantum mechanics of two-dimensional bosons
world lines of bosons in spacetime
t
y
x
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett.
47, 1556 (1981) D.R. Nelson, Phys. Rev. Lett.
60, 1973 (1988) M.P.A. Fisher and D.-H. Lee,
Phys. Rev. B 39, 2756 (1989)
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Boson-vortex duality
Classical statistical mechanics of a dual
three-dimensional superconductor vortices in a
magnetic field
z
y
x
Strength of magnetic field density of bosons
f flux quanta per plaquette
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett.
47, 1556 (1981) D.R. Nelson, Phys. Rev. Lett.
60, 1973 (1988) M.P.A. Fisher and D.-H. Lee,
Phys. Rev. B 39, 2756 (1989)
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Boson-vortex duality
Statistical mechanics of dual superconductor is
invariant under the square lattice space group
Strength of magnetic field density of bosons
f flux quanta per plaquette
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Boson-vortex duality
Hofstädter spectrum of dual superconducting
order
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Boson-vortex duality
Hofstäder spectrum of dual superconducting order
See also X.-G. Wen, Phys. Rev. B 65, 165113
(2002)
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Boson-vortex duality
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Boson-vortex duality
Immediate benefit There is no intermediate
disordered phase with neither order
(or without topological
order).
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Analysis of extended LGW theory of projective
representation
First order transition
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Analysis of extended LGW theory of projective
representation
First order transition
Second order transition
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Analysis of extended LGW theory of projective
representation
Spatial structure of insulators for q2 (f1/2)
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Analysis of extended LGW theory of projective
representation
Spatial structure of insulators for q4 (f1/4 or
3/4)
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B. Application to a short-range pairing model
for the cuprate superconductors
Charge
order and d-wave superconductivity in an
effective theory for the spin S0 sector.
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A convenient derivation of the effective theory
of short-range pairs is provided by the doped
quantum dimer model
Density of holes d
E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B
4, 225 (1990).
27
Duality mapping of doped dimer model shows
(a) Superfluid, insulator, and supersolid ground
states of a theory which obeys the magnetic
algebra
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Duality mapping of doped dimer model shows
(b) At d 0, the ground state is a Mott
insulator with valence-bond-solid (VBS) order.
This associated with f1/2 and the algebra
or
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Duality mapping of doped dimer model shows
(c) At larger d , the ground state is a d-wave
superfluid. The structure of the extended LGW
theory of the competition between superfluid and
solid order is identical to that of bosons on the
square lattice with density f. These bosons can
therefore be viewed as d-wave Cooper pairs of
electrons. The phase diagrams of part (A) can
therefore be applied here.
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Global phase diagram
La2CuO4
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Global phase diagram
g
or
La2CuO4
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989).
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Global phase diagram
g
or
La2CuO4
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Global phase diagram
g
or
La2CuO4
34
Global phase diagram
g
or
La2CuO4
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Spin excitations in a coupled ladder model

• M. Vojta and T. Ulbricht, cond-mat/0402377
• G.S. Uhrig, K.P. Schmidt, and M. Grüninger,
  cond-mat/0402659
• M. Vojta and S. Sachdev, unpublished.

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Bond operator (S. Sachdev and R.N. Bhatt, Phys.
Rev. B 41, 9323 (1990)) theory of coupled-ladder
model, M. Vojta and T. Ulbricht, cond-mat/0402377
J. M. Tranquada et al., Nature 429, 534 (2004)
37
Numerical study of coupled ladder model,
G.S. Uhrig, K.P. Schmidt, and M. Grüninger,
cond-mat/0402659
J. M. Tranquada et al., Nature 429, 534 (2004)
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C. Implications for STM
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Pinning of charge order in a superconductor
Pinning leads to modulations at
wavevectors
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Charge order in a magnetic field
The projective transformations of vortices imply
that each pinned vortex has a halo charge order
modulations this can be computed with no
additional parameters !
41
LDOS of Bi2Sr2CaCu2O8d at 100 K.

M. Vershinin, S. Misra, S. Ono, Y.
Abe, Y. Ando, and A. Yazdani, Science, 303, 1995
(2004).
42
Energy integrated LDOS (between 65 and 150 meV)
of strongly underdoped Bi2Sr2CaCu2O8d at low
temperatures, showing only regions without
superconducting coherence peaks
K. McElroy, D.-H. Lee, J. E. Hoffman, K. M Lang,
E. W. Hudson, H. Eisaki, S. Uchida, J. Lee, J.C.
Davis, cond-mat/0404005.
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Vortex-induced LDOS of Bi2Sr2CaCu2O8d integrated
from 1meV to 12meV
b
J. Hoffman E. W. Hudson, K. M. Lang,
V. Madhavan, S. H. Pan, H. Eisaki, S.
Uchida, and J. C. Davis, Science 295, 466 (2002).
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• Conclusions
• Description of the competition between
  superconductivity and charge order in term of
  defects (vortices). Theory naturally excludes
  disordered phase with no order.
• Vortices carry the quantum numbers of both
  superconductivity and the square lattice space
  group (in a projective representation).
• Pinned vortices are intricately linked to pinned
  charge order.
• Future theory S1/2 and S1 excitations.